A positioning apparatus for positioning a stage, e.g., a wafer stage or reticle stage, used in an exposure apparatus can drive the stage in X, Y, and X translational directions and a rotational direction such as a pitching or rolling direction (see Japanese Patent Laid-Open No. 2000-339032). Generally, the positioning apparatus can drive the stage along the respective drive axes using hydrostatic bearings or electromagnets. The positioning apparatus detects the stage position using a position measurement device, such as a laser interferometer or linear encoder, and supplies a driving force to the stage using a ball screw or an actuator, such as a linear motor, to control the stage position in accordance with positioning servo control or velocity servo control.
With the conventional positioning apparatus, however, when the stage is to be driven along a direction of one degree of freedom, the actuator can generate a thrust in a direction not parallel to the direction of the drive axis. In this case, an undesirable force is generated along other axes. FIGS. 3A and 3B are views for explaining the principle as to how the force is generated along the other axes. FIG. 3A is a front view, and FIG. 3B is a plan view. In the plan view of FIG. 3B, assume that the vertical direction on the drawing corresponds to the X-axis (which is positive downward), and that the horizontal direction on the drawing corresponds to the Y-axis (which is positive rightward). An X stage 21 can be driven in the X direction, and a Y stage 22 can be driven in the Y direction. The X stage 21 is driven by drive actuators 23 and restrained in the Y direction by X stage guides 25. Accordingly, the X stage 21 is driven in the X direction. The Y stage 22 is driven in the Y direction by a Y stage guide (not shown) arranged on the X stage 21.
Each drive actuator 23 generates a thrust F to drive the X stage 21. As shown in FIG. 3B, when the thrust F is inclined with respect to the X-axis, it has a Y-direction-component force. In this case, the thrust F is decomposed into a component Fx and a component Fy with respect to the X- and Y-axes, respectively. At this time, the force component Fy generated in the Y direction is input as a disturbance to the servo control system of the Y stage 22, which is restrained in the X direction by the X stage 21, and degrades the positioning accuracy of the Y stage 22.
FIGS. 4A and 4B are another set of views for explaining the principle as to how the force is generated along the other axes. FIG. 4A is a front view, and FIG. 4B is a plan view. For example, as shown in FIG. 4B, when the drive actuators 23 are not parallel to the X-axis, a displacement X and velocity V generated when the X stage 21 is driven have Y-axis components as well as X-axis components. In this case, the displacement X is decomposed into Xx and Xy with respect to the X- and Y-axes, respectively, and the velocity V is decomposed into Vx and Vy with respect to the X- and Y-axes, respectively. The displacement Xy and velocity Vy generated in the Y-axis are input to the control system of the Y stage 22 as force disturbances by, e.g., the spring performance, damperness, or the like, of actuators 24, which bind the Y stage 22. Consequently, the positioning accuracy of the Y stage 22 is degraded.